Stinson, David W

White-Fredette, Kimberly

English

Georgia State University

Georgia State University
ScholarWorks @ Georgia State University
Middle-Secondary Education and Instructional
Technology Faculty Publications
Department of Middle-Secondary Education and
Instructional Technology (no new uploads as of Jan.
2015)
2007
What is Mathematics?: Teachers Exploring the
Philosophy of Mathematics
Kimberly White-Fredette
David W. Stinson
Georgia State University, dstinson@gsu.edu
Follow this and additional works at: https://scholarworks.gsu.edu/msit_facpub
Part of the Elementary and Middle and Secondary Education Administration Commons,
Instructional Media Design Commons, Junior High, Intermediate, Middle School Education and
Teaching Commons, Mathematics Commons, and the Secondary Education and Teaching
Commons
This Article is brought to you for free and open access by the Department of Middle-Secondary Education and Instructional Technology (no new
uploads as of Jan. 2015) at ScholarWorks @ Georgia State University. It has been accepted for inclusion in Middle-Secondary Education and
Instructional Technology Faculty Publications by an authorized administrator of ScholarWorks @ Georgia State University. For more information,
please contact scholarworks@gsu.edu.
Recommended Citation
White-Fredette, K., & Stinson, D. W. (2007). What is mathematics?: Teachers exploring the philosophy of mathematics [Abstract].
Abstracts and Program of the Annual TEAM-Math Partnership Conference, Pre-session, Tuskegee, AL, 13–16. Available at:
http://team-math.net/tuskegeeconference/presessionabstracts07.html#kwfa Also available at: http://digitalarchive.gsu.edu/
msit_facpub/17/
Title:   What is Mathematics?: Teachers Exploring the Philosophy of Mathematics 
Presenter:   1)   Kimberly White-Fredette, Griffin Regional Education Service Agency 
                     2)   David W. Stinson, College of Education, Georgia State University 
Abstract:  
Background 
 
Visions of effective mathematics teaching and learning are going through tremendous change 
(e.g., see Kilpatrick, Martin, & Schifter, 2003). The National Council of Teachers of 
Mathematics (NCTM) calls for changes to curriculum and evaluation (1989), teaching (1991), 
and assessment (1995). An accumulated summary of these changes is provided within the pages 
of the NCTM’s Principles and Standards for School Mathematics (2000). The Principles and 
Standards describes a vision of mathematics education that is highly ambitious—a mathematics 
education that “requires solid mathematics curricula, competent and knowledgeable teachers 
who can integrate instruction with assessment, education policies that enhance and support 
learning, classrooms with ready access to technology, and a commitment to both equity and 
excellence” (p. 3). In general, constructivist learning, student-centered classrooms, worthwhile 
tasks, and reflective teaching are all a part of NCTM’s vision of mathematics education in the 
21st century (e.g., see Stein, Smith, Henningsen, & Silver, 2000).  
 
To reform mathematics teaching and learning, mathematics educators need to look beyond the 
traditional view of mathematics as fixed and rigid, a subject of absolute truths, what Lerman 
(1990) termed an absolutist view of mathematics. In other words, constructivist teaching and 
inquiry-based learning demands a new view of mathematics, the fallibilist view, which “focuses 
attention on the context and meaning of mathematics for the individual, and on problem-solving 
processes…[and positions] mathematical knowledge…[as a] library of accumulated experience, 
to be drawn upon and used by those who have access to it” (Lerman, 1990, p. 56). Recent studies 
have explored teacher change, examining teacher beliefs and mathematical reform (e.g. see 
Wilson & Goldberg, 1998). This study, however, unlike many previous studies (e.g., see 
Fennema & Nelson, 1997), examined teachers’ philosophies regarding mathematics, not simply 
their beliefs; it sought to understand how teachers philosophically understand mathematics and 
how this understanding affects their philosophies of mathematics teaching and learning and their 
pedagogical practices.  
  
Purpose and Research Questions 
 
The changes within mathematics education recommended by the NCTM are ambitious. How can 
teachers teach children mathematics in ways that are radically different from the ways that they 
were taught—in student-centered classrooms, using investigative problem-solving approaches, 
employing rich mathematical discourse (see Hiebert, 2003, for a discussion of “traditional” 
curricula and pedagogy)? How can teachers challenge traditional beliefs about mathematics as a 
competitive, individualistic subject that they transmit to a “chosen” few students, into a subject 
that is explored with all students (Stinson, 2004)? In general, do teachers ever question the 
philosophical basis of the subject they teach?  
Mathematicians and philosophers, over the past few decades, have challenged society’s 
perception and philosophy of mathematics (e.g., see Davis & Hersh, 1981; Ernest, 1998; Hersh, 
1997; Tymoczko, 1998). These challenges have positioned mathematics as a human activity, an 
activity not based on rote rules and procedures, but guided by intuition, exploration, and 
investigation (Dossey, 1992). But this philosophy is not held consistently among mathematicians 
(e.g. compare Russell, 1919/1993, to Lakatos, 1976), let alone among mathematics teachers and 
teacher educators. The lack of a common philosophy of mathematics has serious ramifications 
for both the practice and teaching of mathematics; it often silences even a discussion of differing 
philosophies (Dossey, 1992). But, without discussing philosophy, can reform truly take hold in 
school mathematics? Ernest (2004) asks mathematics educators to explore five essential 
questions about their subject: What is mathematics? How does mathematics relate to society? 
What is learning mathematics? What is teaching mathematics? What is the status of mathematics 
education as a field of knowledge? These questions challenge educators to not only reflect on 
their pedagogical practices, but also to question their own beliefs about mathematics and 
mathematical teaching and learning.  
This study examined the philosophies of mathematics held among practicing mathematics 
teachers (P-K–College) and describes how those philosophies develop as they participated in a 
graduate-level, seminar that explored the philosophical development of mathematics. Three 
questions guided the study: How can teachers develop an understanding of their personal 
philosophies of mathematics? How do these philosophies change (or not) as teachers explore the 
writings of various philosophers from various traditions of Western mathematics? How do these 
philosophies impact (or not) their philosophies of mathematics teaching and learning and 
pedagogical practices?  
  
Methodology 
 
The participants of the study were 15 graduate students enrolled in an elective, 3-credit hour, 
graduate-level seminar offered at Georgia State University, a large urban research university in 
Atlanta, GA. The 6-week seminar (summer semester 2007) was reading intensive, engaging the 
students in a number of philosophical writings: Davis and Hersh’s (1981) The Mathematical 
Experience, Russell’s (1919/1993) Introduction to Mathematical Philosophy, Lakatos’ (1976) 
Proof and Refutations: The Logic of Mathematical Discovery, Tymoczko’s (1998) New 
Directions in the Philosophy of Mathematics, and Hersh’s (1997) What is Mathematics, Really? 
The focus of the seminar was to challenge teachers’ conceptions of mathematics and to assist 
them in exploring new and different philosophies of Western mathematics and, consequently, 
mathematics teaching and learning. Class sessions were student centered; in that, students 
summarized and presented the seminar readings and facilitated the class discussions.  
The participants include students who were earning a Specialist or Doctor of Philosophy degree 
in mathematics education. Among the participants were elementary, middle, high school, and 
college and university mathematics teachers, school mathematics coaches, and full-time graduate 
students.  
[Note: The study is ongoing.] 
Data collection included an initial, 3–5 page, reflective essay regarding each participant’s 
personal philosophy of mathematics; reading journals from each participant that include written 
summaries of each reading during the seminar, participant-selected significant quotations from 
each reading, and comments regarding the participant’s struggles with each reading and how it 
might (or might not) assist in her or his teaching (and research); a final 8–10 page, reflective, 
academic essay, outlining each participant’s philosophy of mathematics and positioning her or 
his pedagogical practices within that philosophy; and scribed notes taken during each class 
discussion (the scribe notes were a rotating responsibility completed by the students).  
In addition, three participants were selected for two individual, in-depth interviews. The three 
participants were selected purposively (Silverman, 2000) among the students who were 
secondary mathematics teachers. The first interview was a face-to-face, semi-structured, 
traditional, question-and-answer interview (Hollway & Jefferson, 2000), conducted at the end of 
the seminar (last week of July 2007); the second interview was a narrative interview (Hollway & 
Jefferson, 2000), conducted 3 months after the end of the seminar. This interview asked the 
participants to explain how the seminar changed (or not) their pedagogical practices. The 
interviews were recorded and transcribed; narrative analysis (Riessman, 1993) served as the data 
analysis methodology.  
  
Findings and Conclusions 
 
[Note: We will have initial findings and conclusions to report by August 24, 2007.] 
  
References 
 
Davis, P. J., & Hersh, R. (1980). The mathematical experience. Boston: Birkhauser. 
Dossey, J. (1992). The nature of mathematics: Its role and its influence. In D. A. Grouws (Ed.), 
Handbook of research in mathematics teaching and learning (pp. 39–48). New York: 
Macmillan. 
Ernest, P. (1988). The impact of beliefs on the teaching of mathematics. [Electronic Version]. 
Retrieved May 2, 2006 from http://www.people.ex.ac.uk/PErnest/impact.htm 
Ernest, P. (1998). Social constructivism as a philosophy of mathematics. Albany: State 
University of New York Press.  
Ernest, P. (2004). What is the philosophy of mathematics education? [Electronic Version]. 
Philosophy of Mathematics Education Journal, 18. Retrieved January 4, 2006 from 
http://www.people.ex.ac.uk/PErnest/pome18/PhoM_%20for_ICME_04.htm 
Fennema, E., & Nelson, B. (Eds.) (1997) Mathematics teachers in transition.  Mahwah, NJ: 
Lawrence Erlbaum. 
Hersh, R. (1997). What is mathematics, really? New York: Oxford University Press. 
Hiebert, J. (2003). What research says about the NCTM Standards. In J. Kilpatrick, W. G. 
Martin, & D. Schifter (Eds.), A research companion to Principles and Standards for School 
Mathematics (pp. 5–23). Reston, VA: National Council of Teachers of Mathematics.  
Hollway, W., & Jefferson, T. (2000). Doing qualitative research differently: Free association, 
narrative and the interview method. Thousand Oaks, CA: Sage.  
Kilpatrick, J., Martin, W. G., & Schifter, D. (Eds.). (2003). A research companion to principles 
and standards for school mathematics. Reston, VA: National Council of Teachers of 
Mathematics.  
Lakatos, I. (1976). Proofs and refutations: the logic of mathematical discovery. Cambridge, UK: 
Cambridge University Press. 
Lerman, S. (1990). Alternative perspective of the nature of mathematics. British Educational 
Research Journal 16(1), 53–61. 
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for 
school mathematics. Reston, VA: National Council of Teachers of Mathematics. 
National Council of Teachers of Mathematics. (1991). Professional standards for teaching 
mathematics. Reston, VA: National Council of Teachers of Mathematics.  
National Council of Teachers of Mathematics. (1995). Assessment standards for school 
mathematics. Reston, VA: National Council of Teachers of Mathematics.  
National Council of Teachers of Mathematics. (2000). Principles and standards for school 
mathematics. Reston, VA: National Council of Teachers of Mathematics.  
Riessman, C. K. (1993). Narrative analysis (Qualitative research methods: Volume 30). 
Thousand Oaks, CA: Sage Publications. 
Russell, B. (1993). Introduction to mathematical philosophy. New York: Dover. (Original work 
published 1919) 
Silverman, D. (2000). Doing qualitative research: A practical handbook. Thousand Oaks, CA: 
Sage.  
Stein, M. K., Smith, M. S., Henningsen, M., A., & Silver, E. A. (Eds.). (2000). Implementing 
standards-based mathematics instruction: A casebook for professional development. New York: 
Teachers College Press.  
Stinson, D. W. (2004). Mathematics as “gatekeeper” (?): Three theoretical perspectives that aim 
toward empowering all children with a key to the gate. The Mathematics Educator, 14(1), 8–18. 
Tymoczko, T. (Ed.). (1998). New directions in the philosophy of mathematics. Princeton, NJ: 
Princeton University Press.  
Wilson, M., & Goldenberg, M. (1998). Some conceptions are difficult to change: One middle 
school mathematics teacher’s struggle. Journal of Mathematics Teacher Education, 2, 269–293. 
 


What is Mathematics?: Teachers Exploring the Philosophy of Mathematics

ScholarWorks @ Georgia State University

Georgia State UniversityScholarWorks @ Georgia State UniversityMiddle-Secondary Education and InstructionalTechnology Faculty PublicationsDepartment of Middle-Secondary Education andInstructional Technology (no new uploads as of Jan.2015)2007What is Mathematics?: Teachers Exploring thePhilosophy of MathematicsKimberly White-FredetteDavid W. StinsonGeorgia State University, dstinson@gsu.eduFollow this and additional works at: https://scholarworks.gsu.edu/msit_facpubPart of the Elementary and Middle and Secondary Education Administration Commons,Instructional Media Design Commons, Junior High, Intermediate, Middle School Education andTeaching Commons, Mathematics Commons, and the Secondary Education and TeachingCommonsThis Article is brought to you for free and open access by the Department of Middle-Secondary Education and Instructional Technology (no newuploads as of Jan. 2015) at ScholarWorks @ Georgia State University. It has been accepted for inclusion in Middle-Secondary Education andInstructional Technology Faculty Publications by an authorized administrator of ScholarWorks @ Georgia State University. For more information,please contact scholarworks@gsu.edu.Recommended CitationWhite-Fredette, K., & Stinson, D. W. (2007). What is mathematics?: Teachers exploring the philosophy of mathematics [Abstract].Abstracts and Program of the Annual TEAM-Math Partnership Conference, Pre-session, Tuskegee, AL, 13–16. Available at:http://team-math.net/tuskegeeconference/presessionabstracts07.html#kwfa Also available at: http://digitalarchive.gsu.edu/msit_facpub/17/Title:   What is Mathematics?: Teachers Exploring the Philosophy of Mathematics Presenter:   1)   Kimberly White-Fredette, Griffin Regional Education Service Agency                      2)   David W. Stinson, College of Education, Georgia State University Abstract:  Background  Visions of effective mathematics teaching and learning are going through tremendous change (e.g., see Kilpatrick, Martin, & Schifter, 2003). The National Council of Teachers of Mathematics (NCTM) calls for changes to curriculum and evaluation (1989), teaching (1991), and assessment (1995). An accumulated summary of these changes is provided within the pages of the NCTM’s Principles and Standards for School Mathematics (2000). The Principles and Standards describes a vision of mathematics education that is highly ambitious—a mathematics education that “requires solid mathematics curricula, competent and knowledgeable teachers who can integrate instruction with assessment, education policies that enhance and support learning, classrooms with ready access to technology, and a commitment to both equity and excellence” (p. 3). In general, constructivist learning, student-centered classrooms, worthwhile tasks, and reflective teaching are all a part of NCTM’s vision of mathematics education in the 21st century (e.g., see Stein, Smith, Henningsen, & Silver, 2000).   To reform mathematics teaching and learning, mathematics educators need to look beyond the traditional view of mathematics as fixed and rigid, a subject of absolute truths, what Lerman (1990) termed an absolutist view of mathematics. In other words, constructivist teaching and inquiry-based learning demands a new view of mathematics, the fallibilist view, which “focuses attention on the context and meaning of mathematics for the individual, and on problem-solving processes…[and positions] mathematical knowledge…[as a] library of accumulated experience, to be drawn upon and used by those who have access to it” (Lerman, 1990, p. 56). Recent studies have explored teacher change, examining teacher beliefs and mathematical reform (e.g. see Wilson & Goldberg, 1998). This study, however, unlike many previous studies (e.g., see Fennema & Nelson, 1997), examined teachers’ philosophies regarding mathematics, not simply their beliefs; it sought to understand how teachers philosophically understand mathematics and how this understanding affects their philosophies of mathematics teaching and learning and their pedagogical practices.    Purpose and Research Questions  The changes within mathematics education recommended by the NCTM are ambitious. How can teachers teach children mathematics in ways that are radically different from the ways that they were taught—in student-centered classrooms, using investigative problem-solving approaches, employing rich mathematical discourse (see Hiebert, 2003, for a discussion of “traditional” curricula and pedagogy)? How can teachers challenge traditional beliefs about mathematics as a competitive, individualistic subject that they transmit to a “chosen” few students, into a subject that is explored with all students (Stinson, 2004)? In general, do teachers ever question the philosophical basis of the subject they teach?  Mathematicians and philosophers, over the past few decades, have challenged society’s perception and philosophy of mathematics (e.g., see Davis & Hersh, 1981; Ernest, 1998; Hersh, 1997; Tymoczko, 1998). These challenges have positioned mathematics as a human activity, an activity not based on rote rules and procedures, but guided by intuition, exploration, and investigation (Dossey, 1992). But this philosophy is not held consistently among mathematicians (e.g. compare Russell, 1919/1993, to Lakatos, 1976), let alone among mathematics teachers and teacher educators. The lack of a common philosophy of mathematics has serious ramifications for both the practice and teaching of mathematics; it often silences even a discussion of differing philosophies (Dossey, 1992). But, without discussing philosophy, can reform truly take hold in school mathematics? Ernest (2004) asks mathematics educators to explore five essential questions about their subject: What is mathematics? How does mathematics relate to society? What is learning mathematics? What is teaching mathematics? What is the status of mathematics education as a field of knowledge? These questions challenge educators to not only reflect on their pedagogical practices, but also to question their own beliefs about mathematics and mathematical teaching and learning.  This study examined the philosophies of mathematics held among practicing mathematics teachers (P-K–College) and describes how those philosophies develop as they participated in a graduate-level, seminar that explored the philosophical development of mathematics. Three questions guided the study: How can teachers develop an understanding of their personal philosophies of mathematics? How do these philosophies change (or not) as teachers explore the writings of various philosophers from various traditions of Western mathematics? How do these philosophies impact (or not) their philosophies of mathematics teaching and learning and pedagogical practices?    Methodology  The participants of the study were 15 graduate students enrolled in an elective, 3-credit hour, graduate-level seminar offered at Georgia State University, a large urban research university in Atlanta, GA. The 6-week seminar (summer semester 2007) was reading intensive, engaging the students in a number of philosophical writings: Davis and Hersh’s (1981) The Mathematical Experience, Russell’s (1919/1993) Introduction to Mathematical Philosophy, Lakatos’ (1976) Proof and Refutations: The Logic of Mathematical Discovery, Tymoczko’s (1998) New Directions in the Philosophy of Mathematics, and Hersh’s (1997) What is Mathematics, Really? The focus of the seminar was to challenge teachers’ conceptions of mathematics and to assist them in exploring new and different philosophies of Western mathematics and, consequently, mathematics teaching and learning. Class sessions were student centered; in that, students summarized and presented the seminar readings and facilitated the class discussions.  The participants include students who were earning a Specialist or Doctor of Philosophy degree in mathematics education. Among the participants were elementary, middle, high school, and college and university mathematics teachers, school mathematics coaches, and full-time graduate students.  [Note: The study is ongoing.] Data collection included an initial, 3–5 page, reflective essay regarding each participant’s personal philosophy of mathematics; reading journals from each participant that include written summaries of each reading during the seminar, participant-selected significant quotations from each reading, and comments regarding the participant’s struggles with each reading and how it might (or might not) assist in her or his teaching (and research); a final 8–10 page, reflective, academic essay, outlining each participant’s philosophy of mathematics and positioning her or his pedagogical practices within that philosophy; and scribed notes taken during each class discussion (the scribe notes were a rotating responsibility completed by the students).  In addition, three participants were selected for two individual, in-depth interviews. The three participants were selected purposively (Silverman, 2000) among the students who were secondary mathematics teachers. The first interview was a face-to-face, semi-structured, traditional, question-and-answer interview (Hollway & Jefferson, 2000), conducted at the end of the seminar (last week of July 2007); the second interview was a narrative interview (Hollway & Jefferson, 2000), conducted 3 months after the end of the seminar. This interview asked the participants to explain how the seminar changed (or not) their pedagogical practices. The interviews were recorded and transcribed; narrative analysis (Riessman, 1993) served as the data analysis methodology.    Findings and Conclusions  [Note: We will have initial findings and conclusions to report by August 24, 2007.]   References  Davis, P. J., & Hersh, R. (1980). The mathematical experience. Boston: Birkhauser. Dossey, J. (1992). The nature of mathematics: Its role and its influence. In D. A. Grouws (Ed.), Handbook of research in mathematics teaching and learning (pp. 39–48). New York: Macmillan. Ernest, P. (1988). The impact of beliefs on the teaching of mathematics. [Electronic Version]. Retrieved May 2, 2006 from http://www.people.ex.ac.uk/PErnest/impact.htm Ernest, P. (1998). Social constructivism as a philosophy of mathematics. Albany: State University of New York Press.  Ernest, P. (2004). What is the philosophy of mathematics education? [Electronic Version]. Philosophy of Mathematics Education Journal, 18. Retrieved January 4, 2006 from http://www.people.ex.ac.uk/PErnest/pome18/PhoM_%20for_ICME_04.htm Fennema, E., & Nelson, B. (Eds.) (1997) Mathematics teachers in transition.  Mahwah, NJ: Lawrence Erlbaum. Hersh, R. (1997). What is mathematics, really? New York: Oxford University Press. Hiebert, J. (2003). What research says about the NCTM Standards. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to Principles and Standards for School Mathematics (pp. 5–23). Reston, VA: National Council of Teachers of Mathematics.  Hollway, W., & Jefferson, T. (2000). Doing qualitative research differently: Free association, narrative and the interview method. Thousand Oaks, CA: Sage.  Kilpatrick, J., Martin, W. G., & Schifter, D. (Eds.). (2003). A research companion to principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.  Lakatos, I. (1976). Proofs and refutations: the logic of mathematical discovery. Cambridge, UK: Cambridge University Press. Lerman, S. (1990). Alternative perspective of the nature of mathematics. British Educational Research Journal 16(1), 53–61. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics. National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: National Council of Teachers of Mathematics.  National Council of Teachers of Mathematics. (1995). Assessment standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.  National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.  Riessman, C. K. (1993). Narrative analysis (Qualitative research methods: Volume 30). Thousand Oaks, CA: Sage Publications. Russell, B. (1993). Introduction to mathematical philosophy. New York: Dover. (Original work published 1919) Silverman, D. (2000). Doing qualitative research: A practical handbook. Thousand Oaks, CA: Sage.  Stein, M. K., Smith, M. S., Henningsen, M., A., & Silver, E. A. (Eds.). (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York: Teachers College Press.  Stinson, D. W. (2004). Mathematics as “gatekeeper” (?): Three theoretical perspectives that aim toward empowering all children with a key to the gate. The Mathematics Educator, 14(1), 8–18. Tymoczko, T. (Ed.). (1998). New directions in the philosophy of mathematics. Princeton, NJ: Princeton University Press.  Wilson, M., & Goldenberg, M. (1998). Some conceptions are difficult to change: One middle school mathematics teacher’s struggle. Journal of Mathematics Teacher Education, 2, 269–293.  

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